Nullnorm; bounded lattice; direct product

Nullnorms are aggregation functions on the unit interval [0, 1] related to ordinal sums of t-conorms defined on [0, a] and t-norms defined on [a, 1], where a is the annihilator of the considered aggregation function. In this paper some basic algebraic properties like idempotent and nilpotent elements of nullnorms on the unit interval are introduced and discussed. Then, the direct product of nullnorms on bounded lattices is defined and properties of introduced nullnorms are deeply investigated. Thus by defining such algebraic properties and direct product of nullnorms, triangular norms and triangular conorms are extended to a more general form. Finally, we study some properties concerning orders induced by nullnorms acting on lattices.